Knotted toroidal sets, attractors and incompressible surfaces

In this paper we give a complete characterization of those knotted toroidal sets that can be realized as attractors for discrete or continuous dynamical systems globally defined in \({\mathbb {R}}^3\) ${\mathbb{R}}^3$ . We also see that the techniques used to solve this problem can be used to give sufficient conditions to ensure that a wide class of subcompacta of \({\mathbb {R}}^3\) ${\mathbb{R}}^3$ that are attractors for homeomorphisms must also be attractors for flows. In addition we study certain attractor-repeller decompositions of \({\mathbb {S}}^3\) ${\mathbb{S}}^3$ which arise naturally when considering toroidal sets.